## On Exhaustible Resources

Yesterday, George Monbiot wrote in the Guardian that the survival of capitalism relies on persistent economic growth and persistent economic growth is impossible in the long-run because there are finite resources in the world. In response, I made the following popular, but sarcastic tweet.

The tweet was meant to be funny. The format itself is a meme. Nonetheless, it does drive home the point that the source of economic growth is finding more efficient uses of resources. With this being the internet, however, I started receiving replies telling me that I was an idiot who doesn’t understand exhaustible resources and even had one person recommend that I read up on resource economics. As it turns out, I know a little bit about resource economics — and wouldn’t you know it, resource economics actually supports my position. So I thought it was worth a blog post.

Let’s imagine that we have an exhaustible resource. Suppose that the quantity of the exhaustible resource at time $t$ is given by $R(t)$, where $R(0) = R_0 > 0$. Now let’s suppose that $R(t)$ follows a geometric Brownian motion:

$dR = -cR dt + \sigma R dz$

where $c$ is the rate of resource extraction, $\sigma$ is the standard deviation, and $dz$ is an increment of a Wiener process. The intuition of this assumption is as follows. First, zero is an absorbing barrier here. What I mean is that once $R(t) = 0$, it is permanently there. This is the exhaustible resource part. Second, on average the amount of the resource that is available is declining by the consumption of the resource. Third, there is some uncertainty about the quantity of the resource that is actually available. For example, one might observe positive or negative shocks to the supply of the resource. In other words, there are times when new supplies of the resource are discovered. There are other times in which there is less supply than had been estimated. In addition, one could also include “technology shocks” as a source of positive movement in the supply of resources in the sense that better production processes tend to economize on the use of resources, which is basically the same thing as a discovery new amounts of the resource. In short, what we have here is a reasonable representation of how the supply of an exhaustible resource is changing over time.

Now suppose that the consumption of the resource gives us some utility, $u(cR)$ where utility has the usual properties. The objective is to maximize utility over an infinite horizon (with finite resources). Given the process followed by the resources, I can write the Bellman equation for a benevolent social planner as:

$rv(R) = \max\limits_{c} u(cR) - cR v'(R) + \frac{1}{2} \sigma^2 R^2 v''(R)$

where $r$ is the rate of time preference (or the risk-free interest rate). The first-order condition is given as

$u'(cR) = v'(R)$

Intuitively, what this says is that the marginal utility of the consumption of the resource is equal to the marginal value of the resource. Or that marginal benefit equals marginal cost. In fact, this implies that $v'(R)$ is the shadow price of the resource, or the spot price (more on this below).

Now, for simplicity, let’s suppose that consumers have the following utility function:

$u(cR) = \frac{(cR)^{1-\gamma}}{1 - \gamma}$

It is straightforward to show (after A LOT of algebra) that

$c = \frac{r}{\gamma} + \frac{1}{2}\sigma^2 (1 - \gamma)$

So the rate of resource extraction is constant and a function of the parameters of the model. Or, if we assume that there is log-utility, we can simplify this to $c = r.$ Let’s make this further simplification to economize on notation.

So we can re-write our geometric Brownian motion under log utility as

$dR = -rR dt + \sigma R dz$

So now we have the evolution of resources in terms of exogenous parameters. We might be interested in the quantity of resources in existence at any particular point in time, say time $t$. Fortunately, our stochastic differential equation has a solution of the form:

$R(t) = R_0 e^{-[r + (\sigma^2/2)]t + \sigma z(t)}$

Since exponential functions are always positive and $R_0 > 0$, it must be the case that $R(t) > 0, \forall t$.

So what does this mean in English?

What it means is that given the choice about how much to consume of a finite resource over an infinite horizon, the rate of resource exhaustion is chosen to maximize utility. Given the choice of consumption over time, the total supply of the resource will decline on average over time with the rate of resource exhaustion. However, the quantity of the resource will always be positive.

How is this possible?

$u'(cR) = v'(R)$

Recall that I defined $v'(R)$ as the marginal value of the resource, or the shadow price of the resource. Note that as time goes by, $R$ is declining on average. Since $c$ is constant, when $R$ declines, the marginal utility of consumption rises because total consumption $cR$ is declining. It must therefore be the case that shadow price of the resource increases as well. But the problem I described is a planner’s problem (i.e., how a benevolent social planner would allocate the resource given the preferences for society). Nonetheless, a perfectly competitive market for the resource would replicate the planner’s problem. What this means is that as the resource becomes more scarce, the spot price of the resource will rise so that people economize on the use of the resource. Consumption of the resource declines over time such that the resource is never completely exhausted.

Thus, and somewhat ironically given Monbiot’s point, it would be a deviation from competitive markets for the resource or poorly-defined property rights that might lead us to depart from this outcome. So it’s the markets that save us, not the people who want to save us from the markets.

## A Simple Lesson About Money and Models

Imagine you are in your high school algebra class and you are presented with the following two equations:

$x + y = 20$
$2x + 10y = 100$

Two linear equations with 2 unknowns. This is a simple problem to solve.

Now suppose that your teacher gives you the following three equations:

$x + y = 20$
$2x + 10y = 100$
$x + z = 5$

Note that this is still a simple problem to solve. The first two equations are identical to the previous example. You can use those first two equations to solve for x and y. Then, knowing x, you can solve for z. The central point is that the third equation is not important for determining the value of x. The first two equations are sufficient to solve for x and y.

So why am I bringing this up?

This is precisely how the benchmark New Keynesian model deals with money. The baseline New Keynesian model does not include money. The model is complete and a solution exists. Subsequently, to examine whether money would be important in the model, a money demand function is added to this system of equations. There is a solution to the model that exists. Money is then shown to be irrelevant in the determination of the other variables. But, then again, so was z.

UPDATE: I have updated the post to read “benchmark New Keynesian model” to reflect the fact that some have attempted to integrate money into the NK model in other ways, specifically through non-separable utility. This is, in fact, where I am going to take this argument in the future. Nonetheless, for now, see the excellent comment by Jonathan Benchimol below with some links to his related research.

## Macro Musings

This week I was a guest on David Beckworth’s Macro Musings podcast. We discussed my policy brief on the labor standard as well as monetary policy more generally. Here is a link for those interested.

• The topic of this month’s Cato Unbound is J.P. Koning’s proposal for the U.S. to issue a large denomination “supernote” and to tax that note as a way of punishing illegal activity. I will be contributing to the discussion this month along with James McAndrews and Will Luther. You can read J.P.’s lead essay here. The response essays will be linked below the lead essay. My response essay will appear next week.
• My paper with Alex Salter and Brian Albrecht entitled “Preventing Plunder: Military Technology, Capital Accumulation, and Economic Growth” has been accepted at the Journal of Macroeconomics. I think that this paper is based on a really interesting idea (biased, I know). The basic idea is that military technology is a limiting factor for economic growth. We also suggest that both economic growth (at least to some degree) and state capacity could be driven by this common factor.

## Monetary Policy as a Jobs Guarantee

Today, the Mercatus Center published my policy brief on the idea of a “labor standard” for monetary policy that was first proposed by Earl Thompson and David Glasner.

## How Did the Gold Standard Work? Part 1: The Efficiency of the Gold Standard

Sebastian Edwards has written an interesting new book about FDR’s devaluation of the dollar and the legal and economic consequences thereof. This post is not about the book, although I do recommend it. What I would like to write about is motivated by some of the reaction to this book that I’ve seen and heard regarding gold and the gold standard. In recent years, I have become convinced that what I thought was the conventional wisdom on the gold standard is not widely understood. So I’d like to write a series of posts on the gold standard and how it worked. My tentative plan is as follows:

Part 1. The efficiency of the gold standard.
Part 2. The determination of the price level under the gold standard.
Part 3. The Monetary Approach to the Balance of Payments vs. the Price-Specie Flow Mechanism
Part 4. Gold standard interpretations of the Great Depression.

I don’t have a timeline for when these will be posted, but it is my hope to have them posted in a timely fashion so that they can get the appropriate readership. With that being said, let’s get started with Part 1: The efficiency of the gold standard.

Some people define the gold standard in their own particular way. I want to use as broad of a definition as possible. So I will define the gold standard as any monetary system in which the unit of account (e.g. the dollar) is defined as a particular quantity of gold. This definition is broad enough to encapsulate a wide variety of monetary systems, including but not limited to free banking and the pre-war international gold standard. Given this definition, the crucial point is that when the unit of account is defined as a particular quantity of gold, this implies that gold has a particular price in terms of this unit of account. In other words, if the unit of account is the dollar, then all prices are quoted in terms of dollars. The price of gold is no different. However, since the dollar is defined as a particular quantity of gold, this implies that the price of gold is fixed. For example, if the dollar is defined as 1/20 of an ounce of gold, then the price of an ounce of gold is $20. This type of characteristic poses a lot of questions. Does the market accept this price? Or, is there any tendency for the market price of gold to equal the official, or mint price? This is a question of efficiency. If the market price of gold differs substantially from the official price, then the gold standard cannot be thought of as efficient and one must consider the implications thereof for the monetary system. What determines the price level under this sort of system? Does the quantity theory of money hold? What about purchasing power parity? In many ways, these questions are central to understanding not only of how the gold standard worked, but also the nature of business cycles under a gold standard. The price level and purchasing power parity arguments are equilibrium-based arguments. This raises the question as to what mechanisms push us in the direction of equilibrium. We therefore need to compare and contrast the monetary approach to the balance of payments with the price-specie flow mechanism. Finally, given this understanding, I will use the answers of this question to gain some insight into the role of the gold standard with regards to the Great Depression. In terms of efficiency, we can think about the efficiency of the gold market in one of two ways. We could consider the case in which the dollar is the only currency defined in terms of gold. In this case, the U.S. would have an official price of gold, but gold would be sold in international markets and the price of gold in terms of foreign currencies is entirely market-determined. Alternatively, we could consider the case of an international gold standard in which many foreign currencies are defined as a particular quantity of gold. For simplicity, I will use the latter assumption. This first post is concerned with whether or not the gold standard was efficient. So let’s consider the conditions under which the gold standard could be considered efficient. We would say that the gold standard is efficient if there is (a) a tendency for the price of gold to return to its official price, and (b) if market price of gold doesn’t different too much from the official price. Under the assumption that multiple countries define their currency in terms of the quantity of gold, let’s consider a two country example. Suppose that the U.S. defines the dollar as 1/20 of an ounce of gold and the U.K. defines the pound as 1/4 of an ounce of gold. It follows that the price of one ounce of gold is$20 in the U.S. and £4. Note that this implies that $20 should buy £4. Thus, the exchange rate should be$5 per pound. We can use this to discuss important results.

Suppose that the current exchange rate is equal to the official exchange rate. Suppose that I borrow 1 pound at an interest rate $i_{UK}$ for one period and then exchange those pounds for dollars and invest those dollars in some financial instrument in the U.S. that pays me a guaranteed rate of $i_{US}$ for one period.

The cost of my borrowing when we reach the next period is $(1 + i_{UK})\pounds 1$. But remember, I exchanged pounds for dollars in the first period and 1 pound purchased 5 dollars and invested these dollars. So my payoff is $(1+i_{US})\5$. I will earn a profit if I sell the dollars I received from this payoff for pounds, pay off my loan, and have money left over. In other words, consider this from the point of view in period 1. In period 1, I’m borrowing and using my borrowed funds to buy dollars and invest those dollars. In period 2, I receive a payoff in terms of dollars that I sell for pounds to pay off my loan. If there are any pounds left over, then I have made an arbitrage profit. Let $f$ denote the forward exchange rate (the exchange rate in period 2), defined as pounds per dollar. It follows that I can write my potential profit in period 2 as

$(1+i_{US})f\5 - (1+i_{UK})\pounds 1$

We typically assume that in equilibrium, there is no such thing as a perpetual money pump (i.e. we cannot earn a positive rate of return with certainty with an initial investment of zero). This implies that in equilibrium, this scheme is not profitable. Or,

$(1+i_{US})f\5 = (1+i_{UK})\pounds 1$

Re-arranging we get:

$(1+i_{US})f\frac{\5}{\pounds 1} = (1+i_{UK})$

This is the standard interest parity condition. Note that $f$ is defined as pounds per dollar. If the gold standard is efficient we would expect the forward exchange rate to equal to official exchange rate (we should rationally expect that the gold market tends toward equilibrium and the official prices hold). Thus, one should expect that $f = .20$. Plugging this into our no-arbitrage condition implies that:

$i_{US} = i_{UK}$

In other words, the interest rate in both countries should be the same. There is a world interest rate that is determined in international markets.

However, remember that this is an equilibrium condition. Thus, at an point in time there is no guarantee that this condition holds. In fact, in our arbitrage condition, we assumed that there are no transaction costs associated with this sort of opportunity. In addition, under the gold standard, we do not have to exchange dollars for pounds or pounds for dollars. We can exchange dollar or pounds for gold and vice versa. Thus, under the gold standard what we really care about are market prices of the same asset in terms of dollars and pounds. For example, consider a bill of exchange. The price paid for a bill of exchange is the discounted value of the face value of the bill.

$e = \frac{\frac{\pounds 1}{(1 + i_{UK})}}{\frac{\5}{(1+i_{US})}} = \frac{\pounds 1}{\5} \frac{1+i_{US}}{1+i_{UK}}$

So if the ratio of the prices of these bills differ from the official exchange rate, there is a potential for arbitrage profits. As the equation above implies, this can be reflected in the ratio of interest rates. However, it can also simply be observed from the actual prices of the bills.

Officer (1986, p. 1068 – 1069) describes how this was done in practice (referencing import and export points as what we might call an absorbing barrier under which it made sense to engage in arbitrage, given the costs):

In historical fact, however, cable drafts in the pre-World War I period were dominated by demand (also called “sight”) bills as the exchange medium for gold arbitrage. Purchased at a dollar market price in New York, the bill would be redeemed at it pound face value on presentation to the British drawee, with the dollar-sterling exchange rate give by the market price/face value ratio. When this rate was greater than the (demand bill) gold export point, American arbitrageurs (or American agents of British arbitrageurs) would sell demand bills, use the dollars thereby obtained to purchase gold from the U.S. Treasury, ship the gold to London, sell it to the Bank of England, and use part of the proceeds to cover the bills on presentation, with the excess amount constituting profit.

[…]

When the demand bill exchange rate fell below the gold import point, the American arbitrageur would buy demand bills, ship them to Britain, present them to the drawees, use the proceeds to purchase gold from the Bank of England, ship the gold to the United States, and (if not purchased in the form of U.S. coin) convert it to dollars at the U.S. mint.

Thus, we can think of interest rate differentials as opportunities for arbitrage directly, or as reflecting the current market exchange rate. In either case, the potential opportunities for arbitrage profits ultimately kept gold near parity. In fact, Officer (1985, 1986) shows that gold market functioned efficiently (and in accordance with our definition of efficiency).

With this in mind, we have a couple of important questions. This discussion focuses entirely on the microeconomics of the gold standard. In subsequent posts, I will shift the discussion to macroeconomic topics.

## Towards an Alchian-type Approach to Political Economy

In my previous post, I discussed what I called the sleight of hand of an Olson-approach to political economy. The basic idea of that post was that Olson’s theory of concentrated benefits and dispersed costs is often used to malign policies deemed to be inefficient. The sleight of hand aspect is as follows. First, the economist deems a particular policy to be inefficient using a standard theoretical model. Second, the economist hypothesizes that the reason we have such an inefficient policy is due to special interests getting what they want because the costs are dispersed. Third, the economist examines either in historical detail or through regression analysis the role of special interests in getting the policy implemented. Fourth, if special interests are found to have had an effect on the policy being put into place, the economist concludes that the reason we have this inefficient policy is due to special interests. However, the inefficiency of a particular policy is determined by some theoretical model. The empirical finding that special interests had a marginal effect on the policy’s implementation is then used to explain why we got this inefficient policy. Whether or not this is the correct interpretation of the empirical result depends critically on whether the theoretical assertion is true!

Let me elaborate on this point using an example. Consider the example of pollution, which is a principles of microeconomics textbook version of an externality. Suppose that in the absence of special interests, such as environmental groups, pollution would go untaxed. Empirical evidence would show that a tax on pollution was due to the influence of special interest groups (the environmental groups). If we had no concept of externalities and we used the perfectly competitive model as our benchmark model, the conclusion would be that special interests were to blame for this inefficient policy. However, since this is a commonly understood externality, one would not conclude that special interests were to blame for an inefficient policy. On the contrary, the special interest groups would be the reason for implementing the socially optimal policy. In other words, finding evidence of the role of special interest groups does not tell us anything about what is efficient or what is optimal. To do so, we need an explicit theoretical argument or model. Not only that, to come to the correct conclusion we need a correct theoretical model in the sense that it addresses relevant factors, such as externalities.

In my previous post, I criticized economists for too often simply asserting that a policy is inefficient and subsequently applying Olson’s model to explain why we get such stupid, inefficient policies. I also argued that political economy should shift to using an evolutionary argument. In that post, I was short on some of these details. As a result, in this post I want to outline what I meant by this.

In 1950, Armen Alchian published a paper entiled “Uncertainty, Evolution, and Economic Theory.” In that paper he outlines an evolutionary approach to economic theory. Specifically, he argues that it might be misleading to describe firms as profit-maximizers. The reason is that when firms face decisions, there might be a distribution of outcomes across each decision. If these distributions overlap, then it doesn’t make sense to think of the firm as maximizing anything. For example, one distribution might have a higher average profit, but also be associated with a greater variance in profit than some other possible decision. So if firms aren’t maximizing profit, what are they doing?

Alchian suggested that we think of firms as practicing trial-and-error and imitation. Firms try certain things to see what works and imitate things that worked for other firms. Along the way, some firms benefit from good luck and other firms suffer from bad luck. Nonetheless, through this process of trial-and-error, imitation, and uncertainty, the profit mechanism ultimately determines what firms are able to stay in the market and what firms must leave the market. Firms that earn a profit are able to continue operating. Firms that are losing money will be forced to drop out of the market. The profit and loss mechanism therefore selects for firms who are making a profit. This might be due to the firm’s decision-making or it might be due to luck (or some combination of the two). Nonetheless, the economist should be able to explain the success (or lack thereof) of firms. An economist can look at the characteristics and decisions shared by the surviving firms and contrast those characteristics and decisions of the firms that have left the market. In doing so, one can get the sense of the role of decision-making and luck as well as the types of decisions that have proven successful and unsuccessful.

In my view, political economy would benefit from the same sort of approach. Rather than start with a baseline model to determine whether some policy is efficient or not, one should start with the policy itself.

1. What was the primary justification for the policy? More importantly, under what grounds could we consider the policy to be efficient? These questions help to set a much more relevant benchmark than some abstract model of the economist’s choosing.

2. Once these questions have been answered, the economist has some general idea about the conditions under which the policy would be considered efficient. Now, one can take an evolutionary approach to the policy. Did the policy survive for a long time and/or is it still around? What other states or countries have adopted the same or similar policies? How did states and/or countries adopting the policy perform along the relevant dimension in comparison to others? Did any other states/countries have similar policies and abandon them? What happened if they did?

The answers to these questions help to determine the conditions under which the policy survived and the relative success of those places that implemented the policy. This can help to determine if the policy actually achieved what it was supposed to and/or whether the policy is consistent with conditions under which it could be considered efficient. In addition, if the policy seems to have been an efficient response to a particular problem, it is then possible to examine why some places got rid of it and had to bear the cost of doing so thereafter.

In short, I think that it would be useful to do something in political economy with respect to government policy today that Pete Leeson has done with policies and institutions of an earlier time and place. Leeson’s work often starts by examining some policy, law, or institution that seems completely weird, strange, or backwards. He then starts with the premise that it must have been efficient. He outlines the conditions under which the policy or institution would have been efficient and then tests that theory by examining what would have to be true for his efficiency hypothesis to be correct. This approach to political economy or public choice is clearly enlightening, judging by Leeson’s publication record. However, I notice a reticence on the part of many political economists and public choice economists to take the same approach to more recent policies and institutions. The attitude toward more modern policies and institutions seems to be that we “know” that policy X or institution Y is inefficient because economic theory tells us so. Therefore we need to explain why it exists. But how do we “know” this any more than we “know” that trial by battle was a backwards and barbaric practice of no practical use? Leeson’s work showing that trial by battle was a good way of eliciting the true value that particular claimants placed on land was actually quite efficient. So maybe we should be a bit more humble about modern policy as well.